Symposium : Sets Within Geometry : Nancy, France 26-29 July 2011

The Symposium began on the AFTERNOON of Tuesday 26th July and ended on the afternoon of Friday 29th July.

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SCOPE and AIMS OF THE SYMPOSIUM

The Theory of Sets, founded in the last quarter of the 19th Century by Georg Cantor, underwent rapid development at the hands of many contributors.

Within that develoment, several distinct lines can be traced, and these have connected in contrasting ways with the subsequent overall development of mathematical knowledge. The aim of this Symposium is to study and compare these, with particular focus on the question how far the subject matter of set theory can be viewed as part of Geometry

For a more extended discussion of the manner in which this question naturally arises from the study of these various lines of development, see below.

By way of orientation, the organisers of the Symposium offered the following statement of aims :

Those who have come together to organise this Symposium believe that the ultimate aim of foundational efforts is to provide clarifying guidance to teaching and research in mathematics, by concentrating the essential aspects of past such endeavors. By mathematics we mean the investigation of the Relations between Space and Quantity, of the reflected relations between quantity and quantity and between space and space, and the development of our knowledge of these in other words Geometry.

Using tools developed by Cantor and his contemporaries, much more explicit forms of the relation between space and quantity were developed in the 1930s in the field of functional analysis by Stone and Gelfand, partly through the notion of Spectrum (a space corresponding to a given system of quantities). In the 1950s Grothendieck applied those
same tools, around the notion of Spectrum, to algebraic geometry by using and developing the further powerful tool of category theory . Further developments have strongly suggested that it is now possible to incorporate the whole set-theoretic “foundation” of Geometry, explicitly as part of that space-quantity dialectic, in other words as a chapter in an extended Algebraic Geometry.

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Some further Background to the Aims of the Symposium

The Theory of Sets, founded in the last quarter of the 19th Century by Georg Cantor, underwent rapid development at the hands of many contributors and by the mid – 20th Century was regarded by some commentators as providing a framework for the whole of mathematics.

Several distinct lines can be traced within that development, and these have connected in contrasting ways with the subsequent overall development of mathematical knowledge.

In one line of development concepts made explicit by the young Cantor, such as the power set operation and the cohesive/discrete or Mengen/Kardinalen contrast, were extensively used in the algebra, topology, and analysis developed over the last 150 years, resulting in a body of informal methods which might be labelled the *tacit set theory of mathematical practice*. In this tacit theory, as illustrated by Bourbaki’s algebra and topology (but not their official set theory) sets are handled via the kind of isomorphism invariant mapping properties which have long been at the heart of mathematical practice in algebra and geometry

* In the Book of Lawvere and Roseburgh “Sets for Mathematics” it is labelled “Naive Mathematical Set Theory”

In contrast to that line of development and in opposition to features of Cantor’s work. though using others ( his theory of ordinals and his striving for absolute infinity ) there was a line originating in Peano’s dual invention of global singleton and epsilon ( with a sense quite confusingly similar to the local singleton and epsilon made precise by Grothendieck in his 1957 Tohoku paper ). Peano’s aim was to supply mathematical underpinnings for some of
the philosophical positions of Frege. Later Zermelo and von Neumann took up this line of development which became formalized into the theory of the cumulative hierarchy based on a hypothesised relation of absolute and global membership.

That theory is awkward as a framework for mathematical practice, cutting across the requirements of the latter in various respects. Examples abound. As representative instances two will serve :

1) in algebra, to show that a given domain is a ‘subring’ of a field, the image of the injection that naturally arises from the construction must be thrown away and replaced by the original elements.

2) In topology, an important fragment of naïve set theory, involving the lattice of subsets of any given set, needs crucially to be combined with the interactions of these lattices for distinct sets such as the plane and the three-space, but the spurious question of whether one given space is ‘included in’
another (according to the global epsilon relation associated with this “iterative” notion of set), obstructs a focus on the necessary contra-variant and co-variant homomorphisms induced by given transformations between the two spaces.

Such examples indicate why axiomatic set theory in the mould of the epsilon-based formalism has become increasingly detached from other areas of mathematical inquiry .

Over the same period that this detachment was becoming apparent, the definition of the concepts of Category and Functor in 1945 and the subsequent understanding made possible by many co-workers of the functoriality of basic and universal mathematical constructions, in particular the Covariant and Contravariant Functoriality of maps out of or in to their respective domains and co-domains, marked the consolidation and strengthening of a development which could be seen in retrospect to have been under way at least since the work of Galois and Grassmann more than a century earlier. Central to this understanding is the recognition that mathematics is permeated by situations involving adjoint pairs of functors.

One chapter in this development was to make algebraically precise the formerly more loosely connected body of informal methods for which the label “the tacit set theory of mathematical practice” was suggested above. This was achieved by the axiomatic formulation of the Elementary Theory of The Category of Sets by Lawvere in 1963, whereby set theory, by acquiring an explicitly functorial formulation, regained contact with the fundamental feature of constructions in the main areas of mathematical practice

Greatly extending and deepening that re-engagement was the development at the same period of Topos Theory. This arose from Grothendieck’s great work in Algebraic Geometry. Amongst the many facets of its later development has been the recognition, through the work of Lawvere and Tierney, that it provides the natural basis for the study of both variable and constant sets within a framework for which Geometry “supplies the leading aspect” (Lawvere 1973). Indeed one of its earliest conceptual achievements in the work of Tierney and others was to display the content of the then newly discovered epsilon-based forcing techniques used by Cohen to prove the Independence of Cantor’s Continum Hypothesis from the viewpoint of the geometrical meaning of the corresponding topos-theoretic constructions

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In the light of these and other developments, the question naturally arises to what extent the subject matter of set theory should be re-conceived as forming part of Geometry

The Symposium will be devoted chiefly to that question.

Any examination of it involves the recognition that the subject matter of Geometry has itself come to be re-conceived in a way involving a generalisation and progressive deepening of our previous geometric notions, culminating in the work of Grothendieck. In this connection, the remarks in the brief Statement of Aims given above are especially pertinent and worth repeating, as providing clarification of the Title chosen for the Symposium “Sets Within Geometry” :

Using tools developed by Cantor and his contemporaries, much more explicit forms of the relation between space and quantity were developed in the 1930s in the field of functional analysis by Stone and Gelfand, partly through the notion of Spectrum (a space corresponding to a given system of quantities). In the 1950s Grothendieck applied those same tools, elaborated around the notion of Spectrum, to algebraic geometry by using and developing the further powerful tool of category theory . Further developments have strongly suggested that it is now possible to incorporate the whole set-theoretic “foundation” of Geometry, explicitly as part of that space-quantity dialectic, in other words as a chapter in an extended Algebraic Geometry.

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It cannot be stressed too strongly that the claim that Set Theory can, in the above sense, be viewed as grounded in Geometry, does *not* imply – as some have insinuated – a “rejection” of set theory. What is at stake here is rather a *deepening* of set theory and the better understanding of the indispensable role within mathematics of the notions of both constant and variable sets. The object of criticism here is one particular line of development, which appeared to have achieved a hegemonic position, and which rested on an axiomatic fixation of the set concept on the basis of a supposed absolute and global relation of iterated membership.

In this connection the Symposium will also undertake an investigation into contrasting ways of formulating and viewing the Continuum Hypothesis – those determined by the tacit presuppositions generated by membership-based axiomatic set theory as against those ways of viewing the Continuum based on an explicit recognition of the central role of the mapping properties of categories of space as studied in geometry.

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The Topics of the Symposium fall under three broad headings : Mathematics – Conceptual Analysis – History

1. Mathematics

Examination of Mathematical Developments yielding a deeper understanding of the place of sets in mathematics.

See below for further discussion.

2. Conceptual Analysis

The connection between these Mathematical developments and broader analysis of the epistemological sources of mathematical ideas. One illustration of such analysis would be an investigation of the meaning of Extensionality and Choice principles in the setting of Topos Theory

3. History

Related Historical investigations such as a re-examination of the work of
Cantor and Dedekind and other figures in the early development and discussion of different
approaches to the Continuum Hypothesis.

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The following Speakers had hoped and planned to participate but were prevented from doing so

The notion of a stack has its origins in Geometry, more precisely as a tool in the non-abelian cohomology (Giraud 1970) of a Grothendieck topos S. Following a suggestion of Lawvere (Perugia1973), such a notion was introduced and studied (Bunge-Pare 1979) relative to the intrinsic site of all epimorphisms of S. The study of stacks is intrinsically connected with the axiom of choice. The stack completion of an S-category C is constructed (Bunge 1979) as an S-category carved out of the Yoneda embedding of C into the presheaf topos P(C). Examples from cohomology are particularly illuminating. In addition, the stack completion of C represents the so called anafunctors with target C. Completions of his sort, more generally for a closed category V with a faithful functor into Set, and with all limits and colimits, will be called “tight”. We give an equivalent version of the notion of a tight completion in terms of V-distributors (Benabou 1973). The distributors version is particularly suited to exhibit the Cauchy completion (Lawvere 1973) as a tight completion. We clarify the connection between the Karoubi envelope and the Cauchy completion. More precisely, we prove that the axiom of choice (epimorphisms split) holds in V if and only if every Karoubi complete V-category is Cauchy complete. Certain tight completions of a V-category C are what we call here `of Morita type’. Examples are the Cauchy completion in terms of those distributors with target C which have a right adjoint, the (related) completion of essential points of P(C), and the points of the classifying topos B(G) of a groupoid G. Neither the Karoubi envelope nor the stack completion are of Morita type.

Prof. Marta Bunge Talk 2

FOX COMPLETIONS AND LAWVERE DISTRIBUTIONS

The topologist R.H. Fox (1957) introduced a notion of spread in order to
describe branched coverings in topological rather than combinatorial terms. Implicit in
his treatment was a connection between complete spreads and cosheaves, hence with Lawvere
distributions (1983, 1992). This connection was made explicit by J.Funk (1995)
for locales.
A full treatment of the subject for toposes was made possible by the role of the
classifier for Lawvere distributions on a topos X , constructed algebraically by
M.Bunge and A. Carboni (1995). A theory of Fox complete spreads and Lawvere
distributions (or of Singular Coverings of Toposes) is the subject matter of the book by M.
Bunge and J.Funk (2006), in which Kock-Zoberlein monads of the completion type play a
crucial role. In my talk, and after a review of some aspects of this theory, I will list
some open problems.

Click on this Link For A Text of Professor Bunge’s Second Talk (Link not yet activated)

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Prof. Christian Houzel : Title(s) and Abstract(s) to be advised.

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Prof. Anders KOCK

Talk 1 : Monads and Extensive Quantities

Abstract: If $T$ is a commutative monad on a cartesian
closed category, then there exists a natural $T$-bilinear pairing
$T(X)\times T(1)^{X}\to T(1)$ (“integration”), as well as a natural
$T$-bilinear action $T(X)\times T(1)^{X} \to T(X)$. These data
together make the endofunctors $T$ and $T(1)^{(-)}$ (co- and
contravariant, respectively) into
a system of extensive/intensive quantities, in the sense of Lawvere.
A natural monad map from $T$ to a certain monad of distributions (in
the sense of functional analysis (Schwartz)) arises from this integration.

In less Technical terms:

Abstract: If T is a commutative monad on a cartesian
closed category, then there exists a natural T-bilinear pairing from
T(X) times the space of T(1)-valued functions on X (“integration”), as well as a natural
T-bilinear action on T(X) by the space of these functions. These data
together make the endofuncors T and “functions into T(1)” into
a system of extensive/intensive quantities, in the sense of Lawvere.
A natural monad map from T to a certain monad of distributions (in
the sense of functional analysis (Schwartz)) arises from this integration.

Prof. Anders KOCK Talk 2 (To be confirmed)

Title: Geometric algebra of projective lines.

Abstract: The projective line over a local ring carries structure of a
category with a certain correspondence between objects and arrows. We
investigate to what extent the local ring can be reconstructed from the
category.

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Prof. FW Lawvere :

Talk 1 : The Dialectic of Continuous and Discrete in the history of the struggle for

a usable guide to mathematical thought

Dedekind and the young Cantor extracted from the complexity of mathematical cohesions and structures the ideas of structureless lauter Einsen and cardinality, following Steiner’s algebraic geometry. They immediately used them in turn as a base for pure structures such as ordered sets and groups.

Hausdorff and others used them as a base for representing various specific categories of cohesion as consisting also of structures of a slightly different kind.

Moreover the notion of structureless discreteness had also been relativized by Galois and others in the study of algebraic geometry over non-algebraically closed fields. Although practicing mathematicians refer to these collections of lauter Einsen as “sets”, that term is used in another way by students of the Frege-Peano-Zermelo-vonNeumann hierarchy. In order to permit general considerations of these sorts to serve as a useful guide to the development of mathematical thinking, it is necessary to extricate them from the continuing pursuit of the elder Cantor’s idealist speculations about an infinity beyond infinity, which as he himself realized belong more to theology than to science.

See my recent talks

((Bristol 2009) Cantor, Zermelo, & the Category of Categories

(Firenze 2010) Cantor’s lauter Einsen , Galois & Grothendieck

(Pisa 2010) What is a space ?

as well as my 2007 TAC article Axiomatic Cohesion

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Talk 2 : What is a Space?

Abstract : A space is just an object in a category of spaces!

The implied further question is given a very general answer involving lextensivity,

as well as a much more structured answer involving dialectically coupled

(cohesive/discrete) pairs of toposes. Examples can be analyzed and constructed

using the simple geometric paradigm of figures and incidence relations, by which

any lextensive category can be embedded in a Grothendieck topos; more refined

subtoposes of the latter are specified by Grothendieck coverings that embody the

geometrical equivalent of existential/disjunctive conditions on these extended

spaces; a specific example involves a generalization of Maschke means. The

extended spaces always include the Hurewicz exponential spaces, for example,

spaces of functions and distributions equipped automatically with the ambient sort

of cohesion. Examples important for smooth, analytic, and algebraic geometry are

infinitesimally generated, pursuing an observation that goes back to Euler. A smooth

account of points and components for algebraic geometry over a non‐algebraically

closed field is achieved by replacing Cantorian abstract sets with a Galois‐Barr topos

as the discrete aspect. The basic goal is to help make the advances in Algebraic

Geometry during the past 50 years more accessible to students and to colleagues in

related fields by utilizing the simplifying advances in categorical foundations during

the same 50 years, especially guided by proposals made by Grothendieck in 1973

ABSTRACT. I intend to discuss a series of topics related
to foundations and philosophy of mathematics from the perspective
of researcher and teacher.

More detailed Conspectus of Talk :

FOUNDATIONS AS SUPERSTRUCTURE

(The Reflections of a practicing mathematician)

The Content of the idea of Foundations in the wide sense is this: principles of organization of mathematical knowledge and of the interpersonal and transgenerational transferral of this knowledge. When these principles are studied using the tools of mathematics itself, self-referential anxiety raises its ugly head and self-doubts start dominating the autistic psyche of a lonely mathematician …In order to avoid this abyss, one can simply cut the self-referentiality loop, and the way this author did it involved stripping “Foundations” from their normative

functions and to consider various foundational matters simply from the viewpoint of their mathematical content. Then Goedel’s theorem becomes a statement that a certain class of structures is not nitely generated (no big deal but interesting thanks to a new context), and the structures/categories controversy is seen in a much more realistic light: contemporary studies fuse (Bourbaki type) structures and categories freely, naturally and unavoidably, by first structurizing sets of morphisms, then categorifying them, then applying to this vast building principles of homotopy and topology in order to squeeze it down to size etc.

In this way foundations turn into superstructure, and the memory of their foundational provenance is conserved only in the way we are speaking about them.

Observing the nature of changes in Foundations from this perspective, one

sees not so much the replacement of one vision by another but rather a permanent enrichment of intuitions whose creative/pedagogical potential seemed temporarily exhausted. Moreover, the scale of historical legacy and continuity becomes much more visible. Here are some illustrations.

(i) Whatever the fate of the scale of Cantorial cardinal and ordinal infinities, the basic idea of set embodied in Cantor’s famous definition”, as a collection of definite, distinct objects of our thought, is as alive as ever. Thinking about a topological space, a category, a homotopy type, a language or a model, we start with imagining such a collection, or several ones, and continue by adding new types of distinct objects of our thought, whether derivable from the previous ones or new.

(ii) We can decide that we wish to study the category of all projective smooth algebraic varieties and various cohomology functors on it, transcending old fashionedstructures. Then we find out that our ideal goal might be in proving that a universal cohomology functor produces an immense motivic Galois group whose represen

tations solve our initial problem: structures win! (the famous Grothendieck motivic program and its Tannakian embodiment).

(iii) The enrichment may possess its inner logic, and understanding of this logic may be a fascinating challenge even for the creators of new paradigms.

An outstanding example for me is the history of the idea of triangulated categories.

Very briefly, in the categorical development of algebraic topology, at a certain stage one had to produce a framework for treating complexes of (co-)chains of various topological spaces as objects better reecting properties of the space itself than of various ways of choosing these (co-)chains. This led to the complicated definition of triangulated category a la Grothendieck and Verdier. It was successful and influential, until more and more contexts revealed its basic technical flaw (cone is not functorial”). About two decades were required in order to see the remedy that should have been evident to Grothendieck himself from the start. Namely,axiomatizing categories of complexes one should from the start think about morphisms rather than objects. When this was done, categories were created and life became much easier.

To summarize: good metamathematics is good mathematics rather than shack

les on good mathematics.

Yuri I. Manin

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Prof. Jean-Pierre MARQUIS

Talk 1 : Abstract geometric sets and homotopy types

The metaphysics of abstraction :

Abstract :

In this paper, I explore the metaphysical aspects of abstract sets given

via geometric means, namely in the usual terminology, as homotopy 0-

types. I am not claiming that homotopy types should be taken as the

ultimate foundations for mathematics, but rather that homotopy types

exhibit what it means to be abstract for sets in such a foundational frame-

work. I will therefore first rehearse some basic facts about homotopy

types, what we know about them and how we know it. What I want

to emphasize is not so much what homotopy types are, but rather the

type of being they have in the mathematical realm. Once this is clarified,

we can zero in on homotopy 0-types, namely sets. I will compare

and contrast how these sets differ from the ones that are usually thought

of as constituting the foundations of mathematics, e.g. in ZF. Further-

more, as homotopy types already indicates, one cannot apply one and the

same criterion of identity for all mathematical entities. There are inherent

dimensions arising naturally in the abstract geometric framework.

Prof. Jean-Pierre MARQUIS

Talk 2 : Space and sets

The evolution of the notion of topological space in the

First half of the 20th century

(based on joint work with Mathieu BELANGER)

Abstract

In this talk, we explore how mathematicians axiomatized the notion of

space from Hausdorff’s first topological notions to the late 1950’s, that is

just before Grothendieck introduced the notion of topos as a generalized

topological space. In particular, we explore the role played by sets of

points in the various axiomatizations up to Bourbaki and contrast this

role with the apparition and development of the algebraic point of view

introduced by Stone. Whereas in the first case, sets of points are taken

to be ontologically primary and epistemologically prior to any concept of

space, in the latter case, sets of points are derived from the algebraic data

and one can argue that the concept of space becomes prior to the notion

of sets of points. Thus, when Grothendieck comes in, it was possible,

although radical, to reorganize the nature and role of sets of points in the

study of spaces.

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Prof. Colin McLARTY

Talk 1 : Cohomology in the Category of Categories as Foundation

The elimination of the axiom scheme of replacement from the
construction of injective resolutions makes possible a rigorous naive
treatment of cohomological number theory in a natural version of the
Category of Categories axioms.

Prof. Colin McLARTY Talk 2 : To be advised

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Prof. Alberto PERUZZI

Talk 1 : Geometric Roots of Semantics II

Abstract. Formal semantics was mainly done in terms of sets, as extensional
discrete collections apparently needed in order to analyse the logical and
cognitive meaning of any sentence, while intensional semantic theories were
supposed to be inherently non-compositional. In a paper I presented at the
LMPS, X,1995, and published in 2000 with the title “Geometric roots of
semantics”, both the apparent necessity of classical set theory for
semantics and the supposed commitment of intensional theories to
non-compositionality were refuted by making reference to the wider framework provided by topos
theory. At the same time, categorical notions were used to propose a theory
of kinaestethic universal patterns of meaning. Since then, various advances
relevant to ascribe geometry, in a wide sense, the role of setting up the
roots of semantics, call for additions and qualifications of the original
proposal. The paper to be presented in Nancy will elaborate this
subject and will fill some gaps, to the purpose of identifying a basic
theoretical ground of common interest for the foundations of mathematics, logic, and cognitive grammar.

References:

A.P., Geometric Roots of Semantics I: From a Logical Point of View,
in Logic, Methodology and Philosophy of Science X. Abstracts, University of
Florence, Florence 1995, p. 165.

I compare the published Bourbaki’s volume on set theory (the version of 1968) with an unpublished Bourbaki’s draft that treats the same topic in an informal manner, which better reflects the contemporary practice of using sets in mathematics in general and in geometry in particular. Both texts use set-theoretic concepts for developing a theory of mathematical (set-based) structure rather then a set theory for its own sake.

The draft begins with a philosophical introduction explaining the central notion of structure, which is followed by a description of various procedures allowing for obtaining further set-theoretic constructions from a given finite family of base sets. Finally the draft presents a general form of set-theoretic construction called mathematical structure and introduces some operations with structures such as induction, transport and product.

In the published volume the notion of set-theoretic construction reduces to a mere metaphor and gets replaced by a notion of syntactic construction, which is treated in great detail. Crucially, these syntactic constructions represent not informal set-theoretic constructions themselves but certain propositions referring to sets and relations between sets as well as certain relations between such propositions (like the relation of logical inference). This is, of course, a very general feature of the formal axiomatic method, which dates back to Hilbert’s work. Although it is not specific for Bourbaki’s approach comparing the two versions of Bourbaki’s set theory gives an opportunity to evaluate the effect of formalization.

I argue that the Hilbert-style formalization of the informal set theory is not an improvement and hence must be abandoned and replaced by a different theoretical setting based on non-propositional postulates rather than usual axioms (I have in mind here the difference between postulates and axioms in Euclid). In addition to stressing the fact that such a constructive setting better fits the current mathematical practice I explain why the Hilbert-style formalization makes mathematics inapplicable in empirical sciences and show how to fix the problem.

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Professor Lou Crane (KSU)

Title and Abstract to be added.

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Profesor Panagis KARAZERIS (Patras)

Title and Abstract to be added

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CONTRIBUTED TALKS and POSTER SESSIONS

As decribed above, the topics of the Symposium fall under three broad headings :